Form of silicon and method of making the same

ABSTRACT

The invention relates to a new phase of silicon, Si 24 , and a method of making the same. Si 24  has a quasi-direct band gap, with a direct gap value of 1.34 eV and an indirect gap value of 1.3 eV. The invention also relates to a compound of the formula Na 4 Si 24  and a method of making the same. N a 4Si 24  may be used as a precursor to make Si 24 .

STATEMENT OF INTEREST

This invention was made with Government support under Grant NumberW911NF-11-1-0300 awarded by the U.S. Army Contracting Command and GrantNumber DOE DE-SC0001057 awarded by the U.S. Department of Energy. TheU.S. Government has certain rights in the invention.

FIELD OF THE INVENTION

The present invention relates generally to a new form of silicon,precursors that are useful in making a new form of silicon, and methodsof making these compounds. More specifically, the invention is concernedwith Si₂₄, a new form of silicon, and Na₄Si₂₄, a new compound that isuseful as a precursor to make Si₂₄. The invention also relates to othersodium-silicon compounds, including sodium-silicon clathrate compounds,and new methods of making the same.

BACKGROUND OF THE INVENTION

Silicon is the second most common element found in the earth's surface.It has a wide variety of commercial uses, including in electronics andsemiconductors, in metallurgical materials, and in photovoltaics.Silicon has a large impact on the world economy, and much of moderntechnology depends on it.

“Normal” silicon has a diamond structure, d-Si. Despite the prevalenceof “normal” silicon in the photovoltaic industry, silicon is actually arelatively poor absorber of sunlight. This is because d-Si has anindirect band gap of 1.1 eV and a direct bandgap of 3.2 eV. That is,d-Si cannot directly absorb photons with an energy level less than 3.2eV to promote electrons into the conduction band; it requires assistancefrom lattice phonons to transfer momentum of electrons to the conductionband, which are excited by photons with an energy level greater than 1.1eV. This large disparity between the indirect and direct gaps means thatthick layers of silicon are required to absorb light. As a consequence,efficiency decreases and cost increases.

For at least the above reasons, there is a need for new phases ofsilicon with direct or quasi-direct band gaps (nearly degenerateindirect and direct gaps) to improve the light absorption efficiency andlower manufacturing costs. In addition, optically active silicon(silicon capable of readily absorbing and emitting light) is desired inmany optical applications (for example, diodes, lasers, sensors, etc.).

SUMMARY

Broadly stated, the present invention provides a new and improved formof silicon, Si₂₄, and a method of making this compound.

This new phase of silicon (Si₂₄, and also referred to as Si₆, oC24silicon or Cmcm-24 silicon) has a quasi-direct band gap (direct gapvalue is 1.34 eV, indirect gap value is 1.30 eV), making it an excellentcandidate for photovoltaic applications and other optical and electronicapplications.

While formally an indirect gap material, Si₂₄ has a nearly identicaldirect gap (1.30 eV vs. 1.34 eV) and is a much better absorber ofsunlight than d-Si. Therefore, this material will be highly desirable inthe photovoltaic industry. It is also worth noting that the maximumquantum efficiency for a single-junction solar cell is 33.7%(Shockley-Queisser limit) and occurs at a band gap of 1.34 eV, the samevalue for Si₂₄.

The present invention also provides a new compound of the formulaNa₄Si₂₄ (also referred to as NaSi₆) and a method of making the same.Na₄Si₂₄ may be used as a precursor to make Si₂₄.

In one embodiment, the invention relates to a method of producing Si₂₄including the following steps: forming a Na₄Si₂₄ precursor by reacting amixture of silicon and sodium at a pressure between about 7 GPa andabout 15 GPa and a temperature between about 320° C. and about 1500° C.;and subjecting the resultant Na₄Si₂₄ precursor to vacuum conditions at atemperature from about 40° C. to about 500° C. to produce Si₂₄. In anembodiment of the invention, the pressure does not exceed about 12 GPain preparing the precursor so as to minimize product decomposition.However, a pressure up to about 15 GPa may be effective if applied for ashorter period of time.

In one embodiment, the vacuum conditions are maintained below a pressureof about 1×10⁻² torr. In another embodiment, the vacuum conditions areheld below a pressure of about 1×10⁻⁵ torr. However, higher or lowerpressures may be used.

In one embodiment, the resultant Na₄Si₂₄ may be subjected to vacuumconditions for a time ranging from about 1 hour to about 1 day. However,in another embodiment, the resultant Na₄Si₂₄ may be subjected to vacuumconditions for a time ranging from about 1 day to about 10 days.

In one embodiment, the silicon and the sodium used to produce Na₄Si₂₄are in the elemental form. In another embodiment, the silicon and thesodium are in the form of a compound.

In one embodiment, the invention relates to a method of producingNa₄Si₂₄ by reacting a mixture of silicon and sodium at a pressurebetween about 8 GPa and about 12 GPa, and a temperature between about700° C. and about 1000° C.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and areintended to provide further explanation of the invention as claimed.

DESCRIPTION OF THE DRAWINGS

The invention is more fully described by reference to the accompanyingdrawings wherein:

FIG. 1 provides a photograph of an Au/Ta capsule after sample recoveryshowing unreacted capsule material.

FIG. 2 is a p-T domain of applicability of Ta as capsule material.

FIG. 3(a) shows double-stage multianvil assemblies (left) and a highpressure cell (right) used for synthesis.

FIG. 3(b) provides photographic images of metallic Na₈Si₄₆ (left) andNa₄Si₂₄ (right) synthesized at high pressure.

FIG. 4 provides dimensional X-ray diffraction data for pieces of Na₈Si₄₆and Na₄Si₂₄.

FIG. 5 provides Rietveld refinements of Powder X-ray diffraction (PXRD)patterns for Na₈Si₄₆ crystallized under slow cooling at 6 GPa. Acorresponding structure is shown to the right with host silicon andguest sodium atoms.

FIG. 6(a) provides a Powder X-Ray diffraction pattern of the sII/d-sisamples obtained in quenching experiments at 6 GPa and 700 C (PL 705a).The weak amorphous halo at ˜10° (2θ) arises from the borosilicate glasscapillary.

FIG. 6(b) provides a crystal structure of Na—Si sII clathrate.

FIG. 7 provides a Rietveld refinement of PXRD patterns for Na₄Si₂₄crystallized under slow cooling at 8 GPa. A corresponding structure isshown to the right with host silicon and guest sodium atoms.

FIGS. 8(a) and 8(b) provide structural fragments of Na₄Si₂₄.

FIG. 9 provides a graph showing electrical resistivity of Na₄Si₂₄ andNa₈Si₄₆ crystallized at 3 GPa as a function of temperature. The graphalso provides a quadratic fit of Na₄Si₂₄ resistivity data (above 75 K).The inset shows the calculated band structure of Na₄Si₂₄ at 1 MPa.

FIG. 10 provides a graph showing the calculated ΔH_(f) of formation(Na_(x)Si_(y)=yd-Si+x Na) for various Na—Si compounds at OK andpressures up to 15 GPa.

FIG. 11 provides a graph showing calculated enthalpies of clathratephases relative to the Na₄Si₄—Si system (15 at. % Na) as a function ofpressure at OK (lines). Shaded boxes at the bottom show the experimentalpressures of d-Si, sI and Na₄Si₂₄ crystal formation at high temperaturesfrom Na—Si melt.

FIG. 12 provides structure determination and composition information forSi₂₄. FIG. 12A provides PXRD data (points) and Rietveld refinements(upper lines) for Na₄Si₂₄ (top) and Si₂₄ (bottom). Lower lines indicatedifference between experimental data and refinement. The upper tickmarks indicate reflection positions for Na₄Si₂₄ (top) and Si₂₄ (bottom).The lower tick marks indicate reflection positions for d-Si (1.5 wt %impurity in top panel). FIG. 12B provides EDXS spectrum obtained fromiridium coated Si₂₄ sample. No sodium is detected (position of redarrows). Carbon and oxygen originate from organic contamination and anative oxide layer, respectively. Inset shows SEM/EDXS mapping image of˜5 micron crystals. Purple color indicates silicon. FIG. 12C shows theeffect of sodium removal on lattice constants and table of measured(EXP) and calculated (DFT) lattice parameters for Na₄Si₂₄ and Si₂₄.

FIG. 13 shows a plot of electrical resistance as a function oftemperature for Si₂₄ an dis fit to extract a band gap of 1.3 eV.

FIG. 14 provides the absorption (imaginary portion of the dielectricconstant) for Si₂₄ compared with “normal” diamond phase silicon.Enhanced light absorption was observed compared with normal silicon.

FIG. 15 provides crystal structures of Na₄Si₂₄ and Si₂₄. FIG. 15Aprovides a schematic of compositional change from Na₄Si₂₄ (left) to Si₂₄(right). Na atoms are shown in purple and silicon in yellow. FIG. 15Bshows an Si₂₄ unit cell exhibiting three crystallographically uniquepositions in each color. FIG. 15C provides a fractional view of Si₂₄emphasizing its channel structure. Channels are formed by eight-memberrings along the a-axis, which are linked by six-membered rings on thetop and sides. These channels are connected along the c-axis byfive-membered rings.

FIG. 16 provides electronic and optical properties of Si₂₄. FIG. 16Ashows calculated Si₂₄ band structure (PBE+G₀W₀). FIG. 16B provides azoomed in region of band gap. Arrows indicate direct (E_(d)) andindirect (E_(i)) gaps. FIG. 16C shows electrical conductivity of Na₄Si₂₄and Si₂₄ (inset shows fit of intrinsic conductivity region with band gapof 1.3 eV). FIG. 16D provides Tauc plots of Kubelka-Munk absorption(K/S) for Si₂₄ obtained from optical reflectivity measurements.Absorption edges are observed at 1.29 eV and 1.39 eV assuming indirectand direct electronic transitions, respectively.

FIG. 17 depicts a unit cell of Si₂₄ and unique silicon atoms. Threedifferent atomic positions of Si are represented with different colors.Crystal fragments are shown on the right side with the most deviatedangle from the perfect tetrahedral angle (109.5°). Angles are from DFToptimization.

FIG. 18 is a graph of lattice parameters of Na_(x)Si₂₄ with respect tosodium concentration x.

FIG. 19 shows the electronic density of state for Na_(x)Si₂₄.

FIG. 20 shows phonon dispersion relations of Si₂₄ at ambient pressure(a) and 10 GPa (b).

FIG. 21 provides a raman spectrum of Si₂₄ with experimental data(bottom) and calculated data (top). Calculated spectra are representedby Lorentzian peaks. The symmetry of each mode is indicated.

FIG. 22 shows the difference between E_(g) and E_(i) during uniaxialcompression along c.

DETAILED DISCUSSION

Applicants first describe certain silicon clathrates and new andunexpected methods of making such compounds.

Silicon clathrates (1) (or simply clathrates throughout thisapplication) are analogues to the water-based clathrate hydrates, (2)often crystallizing as high-symmetry cubic structures. Si atoms withinthese compounds take on an sp³ hybridization state to form a covalentnetwork of polyhedral cavities that host appropriately sized “guest”atoms. The alkali and alkaline earth metals can be intercalated intothese cages that extend in three dimensions. The cubic structure I (sI,Pm3n, a ˜10.2 Δ) clathrate contains 2 small 5¹² cages (12 pentagonalfaces) and 6 larger 5¹²6² cages (12 pentagonal faces and 2 hexagonalfaces) with 46 silicon atoms per unit cell, whereas a unit cell of cubicstructure II (sII, Fd3m, a ˜14.7 Å) is composed of 16 5¹² cages and 8large 5¹²6⁴ cages with 136 Si atoms. The rigid covalent 3-D network andintercalation nature of these compounds give rise to a combination ofinteresting properties, including remarkable incompressibility (3, 4);glasslike thermal conductivity (5); metallic, semi- (6), andsuperconductivity (7, 8); photovoltaic properties (6); and potential forhydrogen storage materials (9).

Most known silicon clathrates have been synthesized at ambient pressureor under deep vacuum using ZintI compounds as precursors or throughdifferent types of chemical reactions (1, 6) Up to now, chemicaldecomposition (arc-remelting or heating in vacuum/Ar atmosphere) remainsthe principal method for Na—Si clathrate synthesis (6, 10, 11). Later,chemical oxidation was proposed as an alternative route (12), as well asspark plasma treatment of the Na₄Si₄ precursor. Although single crystalsof some Si clathrates can be grown by the techniques just mentioned (10,13), the equilibrium synthesis from a melt remains a challenging taskfor well-controlled crystal growth. The knowledge of thermodynamicstability domains in terms of composition, temperature, and pressure is,therefore, of primary importance.

The generalized thermodynamic stabilities of silicon clathrate phasesare not clear. Guest-free Si clathrates have been predicted to be stableonly at negative pressures below −3 GPa (14, 15), (a direct analogy maybe drawn between the negative pressure stability of empty silicon andwater-based clathrates). While some Ba- and halogen-based siliconclathrates can be synthesized at high pressure (16, 17), ab initioresults on the high-pressure stability of Na—Si clathrates suggest thatthese phases are thermodynamically unstable with respect todecomposition at high-pressure conditions (10), Thus, the p-T domainsfor thermodynamic stability and the possibility for equilibrium crystalgrowth need clarification.

At present, the phase equilibria for the Na—Si system remain the leastunderstood as compared with other clathrate systems (e.g., Ba—Si).Sodium is the lightest metal that is known to intercalate the Si cagesalone, and its understanding may lead, for example, to the synthesis ofLi-, Mg-, and Be-bearing clathrates. The experimentally establishedbinary phase diagram at ambient pressure contains only one congruentlymelting compound of Na and Si, sodium silicide (Na₄Si₄), and clathrateformation has not been observed under (quasi-) equilibrium conditions(18). In situ study of Na—Si clathrate growth under high-temperature,high-vacuum conditions allows one to conclude that its formation isprincipally determined by mutual structural relationships of phases anddynamic phenomena (e.g., sodium volatility), rather than thermodynamicstability of corresponding phases (19).

Contrary to Ba-based clathrates (20), high-pressure synthesis has notbeen explored so far for the Na—Si system. The remarkable negativevalues of atomic ΔV of clathrate formations (1.5-2.5 cm³/mol; Table 1,below), compared with the elemental constituents, suggests that highpressure should facilitate their formation. High pressure (HP) synthesisis the most reliable method for elaboration and properties control ofhigh-pressure phases, such as diamond (21, 22), cubic boron nitride (23,24), and boron-rich compounds (25). The quasi-equilibrium growth of HPphases from solvents allows production of high-purity single crystals ofa given habit and high-quality powders.

TABLE 1 Mean Atomic Volumes of Formation V_(at) °_(f) of sodiumclathrates (cm³/mol) Clathrate <V_(at)>^(clathrate) <V_(at)>^(Na+Si)ΔV_(at)°_(f) Na₂₄Si₁₃₆ 12.03 13.81 −1.78 Na₈Si₄₆ 11.83 13.79 −1.96 NaSi₆11.35 13.72 −2.37

This application discloses experimental and theoretical results thatunambiguously indicate that Na clathrates are thermodynamically stablehigh-pressure phases. Stoichiometric sodium clathrates of structure I(Na₈Si₄₆) and a new metallic clathrate compound Na₄Si₂₄ were synthesizeddirectly from the elements, allowing for new opportunities formelt-based growth under equilibrium conditions. Over the range ofconditions studied, the Na₂₄Si₁₃₆ clathrate (sII) only forms as anintermediate compound prior to crystallization of the sI phase.

Sodium clathrates were prepared from elemental Na/Si mixtures (20 mol %Na, i.e., 5% excess as compared with stoichiometric sI and sIIclathrates). The initial mixture was ground for 1 hour in a porcelainmortar contained within a glovebox under a high-purity Ar atmosphere.The mixtures were loaded into double-walled capsules, the inner one fromTa and the outer one from Au (see FIG. 1). Such a simple design of thecapsule allows one to keep the Na-rich liquid under high-pressure andhigh-temperature conditions for a long time (at least up to 12 hours)without hermetic sealing of the initial capsule. The capsules were thenintroduced into high-pressure cells: (1) octahedral-shaped assembliesfor multianvil (18/11 type, pressures from 3 to 8 GPa, FIGS. 1 and 3a)(26) and (2) standard half-inch assemblies for piston-cylinder(pressures from 1 to 3 GPa), and compressed to the desired pressure. Thetime-pressure-temperature conditions of experiments are given in Table 2(see below). In some cases, the heating was performed in two steps,i.e., 30 min at 675 K, and the remaining time (1.5-24 hours) at thefinal temperature. The main purpose of this preheating was to avoid theblow-out of overheated liquid sodium from the cell at the initial stagesof synthesis, that is, before Na reacts with Si. After completion ofthis step, the temperature was decreased either by switching off thepower or by slowly decreasing the power over a 10-60 min duration. Thelatter allowed for the study of crystallization in (quasi-) equilibriumconditions.

TABLE 2 Experimental conditions of High Pressure synthesis and phasecomposition of the resulting washed sample Phase compositionExperimental Ta or Run conditions Si sI SII Na₄Si₂₄ Au Ta_(x)Si_(y)Quenching experiments PL695 6 GPa, 675/1075 K S M W 0.5/1.5 h PL702 6GPa, 675/1075 K S M M 0.5/3.5 h PL703 6 GPa, 675/1075 K S M M 0.5/1.5 hPL704 3 GPa, 675/1075 K W W S 0.5/11.5 h PL705a 6 GPa, 675/975 K S M0.5/3.5 h Slow-cooling experiments PL706 6 GPa, 675/1125 K W S 0.5/5.5 hPL708 3 GPa, 675/975 K W S 0.5/5.5 h PL713 8 GPa, 675/1075 K W S 0.5/5.5h SamiPC1 1 GPa, 1075 K S M 24 h PC1015 2 GPa, 1125 K M W 6 h(33at % Na)PC1028 1.2 GPa, 875 K M 2 h(33at % Na) PC1037 2 GPa, 925 K, S M M 2 hPC1047b 2 GPa, 1025 K, S M 2 h Direct transformations. PC1047a 2 GPa,1025 K S M 2 h(Na₈Si₄₆) * If not mentioned otherwise, the initial Na/Simixtures contained 20 at % of Na. †S (strong) - predominant phase, M(medium) - one of major phases, W (weak) - secondaryphase/contamination. ‡ The format is: Preheating temperature (ortime)/Heating temperature (or time).

As shown in FIG. 1, the recovered samples were easily removed from theTa/Au capsules. No reaction between the Na/Si mixture and capsule wasdetected to 1275 K at 6 GPa and to 1000 K at 3 GPa (see FIG. 2).Recovered samples were washed in water, rinsed with ethanol, and driedin air. FIG. 3(b) shows typical pictures of recovered clathrateparticles with metallic brilliance, similar to previous reports forstoichiometric clathrates (10).

Powder X-ray diffraction (PXRD) from samples recovered between 2 and 6GPa revealed the formation of sI clathrate. The presence of a smallamount of silicon in the recovered samples likely indicates theperitectic character of the Na—Si phase diagram under pressure, with theperitectic reaction L_(Na/Si)+Si=Na₈Si₄₆. Two dimensional X-Raydiffraction data indicated the presence of both primary grown clathratecrystals and fine-grained powder formed during eutectic crystallization(see FIG. 4). Individual polycrystalline pieces of pure sI clathratewere easily isolated (up to 500 μm in length, FIG. 3), and FIG. 5 showsthe PXRD pattern of sI clathrate formed under quasi-equilibriumconditions at 6 GPa with a=10.208(1) Å. Correspondence between theexperimental lattice constants and literature values (a=10.198 Å) (27),in addition to Rietveld refinements of site occupancies (Table S3,below), indicate that the recovered phase is approximatelystoichiometric, that is, Na₈Si₄₆. The formation of sI clathrate directlyfrom the elements indicates a region of high-pressure thermodynamicstability on the Na—Si binary phase diagram.

TABLE S3 Structural refinement data for Na₈Si₄₆ and NaSi₆. Differentfits for a given structure correspond either to the model of idealstoichiometry (Fit 1) or to possible non-stoichiometry of Na positions(Fit 2). Both models provide comparable fits indicating near-idealstoichiometry. A decrease in Na occupancy leads to a decrease in thefitted U_(iso) value. Na₈Si₄₆ Fit 1 X² = 0.260 · 10⁻² wRp = 0.0449(-Bknd) Rp = 0.0305 (-Bknd) Space group: Pm-3n (No 223) Latticeparameter: a = 10.208(1) Atomic coordinates x y z Occupancy U_(iso) Si1(6c) 0.25 0.5 0 1 0.012(6) Si2 (16i) 0.184(1) 0.184(1) 0.184(1) 10.012(6) Si3 (24k) 0 0.118(1) 0.308(1) 1 0.012(6) Na4 (2a) 0 0 0 10.042(30) Na5 (6d) 0.25 0 0.5 1 0.051(20) Fit 2 X² = 0.257 · 10⁻² wRp =0.0443 (-Bknd) Rp = 0.0289 (-Bknd) Space group: Pm-3n (No 223) Latticeparameter; a = 10.208(1) Atomic coordinates x y z Occupancy U_(iso) Si1(6c) 0.25 0.5 0 1 0.012(6) Si2 (16i) 0.184(1) 0.184(1) 0.184(1) 10.012(6) Si3 (24k) 0 0.118(1) 0.308(1) 1 0.012(6) Na4 (2a) 0 0 0 1.0(1)0.043(30) Na5 (6d) 0.25 0 0.5 0.95(6) 0.032(30) NaSi₆ Fit 1 X² = 0.866 ·10⁻² wRp = 0.0322 (-Bknd) Rp = 0.0500 (-Bknd) Space group: Cmcm (No 63)Lattice parameters: a = 4.106(3), b = 10.563(9), c = 12.243(9) Atomiccoordinates x y z Occupancy U_(iso) Na1 (4c) 0 0.285(5) 0.25 1 0.061(20)Si2 (8f) 0 0.245(3) 0.549(1) 1 0.017(6) Si3 (8f) 0 0.566(2) 0.347(1) 10.017(6) Si4 (8f) 0 0.027(2) 0.594(1) 1 0.017(6) Fit 2 X² = 0.845 · 10⁻²wRp = 0.0327 (-Bknd) Rp = 0.0508 (-Bknd) Space group: Cmcm (No 63)Lattice parameters: a = 4.106(3), b = 10.563(9), c = 12.243(9) Atomiccoordinates x y z Occupancy U_(iso) Na1 (4c) 0 0.284(4) 0.25 0.93(6)0.043(30) Si2 (8f) 0 0.245(3) 0.549(1) 1 0.018(6) Si3 (8f) 0 0.565(2)0.346(2) 1 0.018(6) Si4 (8f) 0 0.028(2) 0.594(1) 1 0.018(6)

While sI was always observed between 2 and 6 GPa under quasi-equilibriumconditions of slow cooling (2-10 K·min-1), the rapid quench of thesystem to ambient temperature (by abruptly switching off the heatingpower) led to the recovery of both sI and sII phases. Fast and/or lowertemperature treatment of the reacting mixture may also occasionally leadto a combination of sII and diamond-Si, without remarkable traces of sIclathrate (see FIGS. 6(a) and 6(b)). According to X-ray diffractiondata, the lattice parameter of this sII clathrate (a=14.76(2) A) isexpanded by 0.3% as compared to that of stoichiometric sII obtained bythermal decomposition under vacuum (a=14.72 Å) (28). Evidently, sII isnot thermodynamically stable between 2 and 6 GPa over the temperaturerange studied, but its formation may be induced under transientrecrystallization conditions. This result might be explained in terms ofOstwald's rule of stages (i.e., easier nucleation of a less stable phasein terms of chemical potential, prior to the stable phase formation),due to structural similarity and close Gibbs energies. A similar HPeffect was recently observed in other cage-based crystal structures ofelemental boron (29).

When the pressure was increased to 8 GPa, the formation of a novel Na—Sicompound, Na₄Si₂₄, was observed. As shown in FIG. 7, it has theEu₄Ga₈Ge₁₆(30) structural type, never reported thus far for an alkalimetal. The structure is composed of sp^(a)-bonded Si atoms, which formtunnels, intercalating Na atoms along the a axis. The Na˜Na distance of4.106(1) Å is the shortest metal-metal distance for this structuralfamily (e.g., Ba˜Ba distance in isostructural BaSi₂₄ is 4.479 Å), butquite reasonable given the small diameter for Na. The Si—Si distancesvary from 2.368 to 2.380 Å, very close to those of sI clathrate Na₈Si₄₆(2.333-2.413 Å), and shorter than those in BaSi₂₄ (2.400-2.469 Å). TheSi—Si—Si angles vary in a wide range between 99.2 and 134.0°, similar tothose in BaSi₂₄₎(97.8-136.4°, and are remarkably less uniform than thosein cubic sI Na₈Si₄₆₎(106.1-124.1°. As shown in FIG. 8(a), each sodiumatom has 14 nearest Si atoms (3.197-3.436 Å) and 2 Na atoms along the aaxis. Contrary to sI and sII clathrate structures, where Si atoms formcages produced by only slightly distorted pentagonal and hexagonalrings, the Si matrix of Na₄Si₂₄ is produced by hexagonal rings withwell-pronounced boat and chair configurations (see FIG. 8(b)).

As compared to Ba₄Si₂₄(31), Sr₄Si₂₄(32) and Ca₄Si₂₄(33), thecorresponding sodium compound forms at substantially lower pressure: 8GPa as compared to 11.5 and 10 GPa for BaSi₂₄ and CaSi₂₄, respectively.The pressure for Na₄Si₂₄ formation allows consideration of this phasefor a large-volume production, for example, in the toroid-typehigh-pressure systems, (34) contrary to the alkali-earth metalanalogues.

The electrical resistivity of a polycrystalline piece of 500×500 μm² ofNa₄Si₂₄ was measured as a function of temperature by the standardfour-electrode technique, using a Physical Property Measurement System(PPMS: Quantum Design, Inc.). Overall, Na₄Si₂₄ exhibits metallicbehavior, with the electrical resistivity decreasing with decreasingtemperature (see FIG. 9). Below 75 K, the electrical resistivity beginsto increase with further temperature decreases, a feature never observedso far for related compounds within this structural family. The residualresistivity ratio value (RRR, defined here as ratio between ρ_(300K) anda minimal value of ρ, ρ_(min)) is between 13 and 100 depending on theestimate of ρ_(min), which can be taken at 75 K (for the lower estimateof RRR) or taken at 0 K by extrapolating a least-squares fit of thehigh-temperature part of the T-ρ curve (FIG. 9). Contrary to Na₄Si₂₄,the resistivity of sI obtained at 3 GPa shows monotonic behavior withtemperature (FIG. 9), but the RRR value is also very high, 24. TypicalRRR values for clathrates obtained without pressure are between 1 and 6,even for single crystals (35) (the best value of 36 was recentlyachieved (36) for a single crystal of Na₈Si₄₆ obtained by the so-called“slow controlled removal of Na” method) (37). In fact, high-pressuresynthesis provides samples with remarkably improved electricalproperties, even in the case of bulk material. Until now, such animprovement was unachievable by nonequilibrium crystal growth methods.

The observed minimum at about 75 K on the resistivity curve for Na₄Si₂₄might be caused by the particularities of the electronic band structureand by an unknown scattering process. The present results suggest thatthe resistivity increase in Na₄Si₂₄ at very low temperatures (formalvalue of ρ_(300K)ρ_(2K) for Na₄Si₂₄ is 6) is not due to structuraldefects, intergrain boundaries, or an experimental artifact (thelow-temperature resistivity curve of sI does not show such behavior,FIG. 9).

While the formation of sI clathrate and Na₄Si₂₄ directly from theelements indicates high-pressure thermodynamic stability of thesephases, the inventors performed density functional theory (DFT)calculations (38) in order to further elucidate the nature of theirstability. Geometry optimizations were performed within the framework ofthe generalized gradient approximation (GGA) with thePerdew-Burke-Ernzerhof (PBE) parametrization (39) for theexchange-correlation functional implemented in Quantum Espresso. ForBrillouin zone integration, the inventors used the Monkhorst-Pack scheme(40) and checked convergence of ground-state calculations with uniformlyincreasing k-point meshes for each structure. The inventors used aplane-wave basis set cutoff of 60 Ry and generated a 8×8×8 k-point gridmeshed for a Brillouin zone integration (41). Calculations of phonons ofNa₄Si₂₄ were performed with density functional perturbation theory witha uniform 4×4×4 mesh.

Density functional theory (DFT) calculations reveal that Na₄Si₄, sI,sII, and Na₄Si₂₄ are all stable against decomposition into the elementsat high-pressure conditions (see FIG. 10). FIG. 10 shows the calculatedenthalpy differences, ΔH, between the clathrate structures and a mixtureof d-Si and Na₄Si₄ Zintl phase, which are the experimentally observedthermodynamically stable phases at ambient conditions (18). To comparethe thermodynamic stability, the total composition has been fixed to 15at. %; that is, the inventors compared the systems NaSi_(5.667) forstructure II, (NaSi_(5.75-0.083) Si) for structure I, and (Na₄Si₂₄-0.333Si) for Na₄Si₂₄. As shown in FIG. 11, these results indicate that bothsI and Na₄Si₂₄ are indeed thermodynamically stabilized by high-pressureconditions: sI and Na₄Si₂₄ become the most enthalpically favorablestructures at approximately 4 and 9.5 GPa, respectively. From theexperimental high pressure, high temperature (HPHT) data, thecorresponding values of pressure are 2.5(5) GPa for sI and 7(1) GPa forNa₄Si₂₄ at 900-1100 K, demonstrating very good qualitative agreementbetween predicted and experimental domains of stability.

Density functional theory calculations suggest that sII clathrate is thestable ground state from 0 to 4 GPa, yet experiments performed below 2GPa resulted in the recrystallization of Si. At ambient pressure, theinventors calculate sII to be more stable than Na₄Si₄+Si by 2.5meV/atom, whereas sII is more stable than “sI-Si” by 5.2 meV/atom. At4.2 GPa, the inventors calculate sI to become more stable than sII. Theabsence of sII in quasi-equilibrium experiments and the lack of thisclathrate phase on the ambient-pressure binary Na—Si phase diagram arein apparent disagreement with our calculations; however, enthalpydifferences between these compounds are small. Finite temperatureeffects and kinetic barriers across the various phases may contribute tothe differences observed. Nevertheless, the observation of sII inrapid-quench experiments certainly verifies the seemingly isoenthalpicnature of these phases.

The results open new perspectives for high-pressure synthesis andproperties control of new advanced materials. The high-pressurethermodynamic stability of Na—Si clathrate phases allows for amelt-based synthesis approach, which could be very useful forcompositional control in mixed phases (e.g., Na+Ba, etc.), high-qualitysingle crystals, and for precise tuning of the occupancy ratios. Allphases formed in this pressure domain allow for larger-volume scaling ofmaterials (from 40 cm³ for cubic sI at 3 GPa to 1 cm³ for theorthorhombic Eu₄Ga₈Ge₁₆ structure at 8 GPa). The consistency ofexperimental results with ab initio calculations may justify the futureapplication of this approach to the prediction of new covalent sp^(a)intercalation compounds (e.g., carbon-rich compounds). Finally, theresults reveal the existence of multiple chemical mechanisms that allowfor synthesis of high-pressure phases “without pressure.” Since theNa—Si clathrates are stable only under high-pressure conditions (>2GPa), previous reports of their synthesis may be viewed asnonequilibrium, precursor-based routes to high-pressure phases atlow-pressure conditions. The understanding of such intrinsicinterrelationships between thermodynamics and kinetics is thus the nextstep to explore that could open the potential for other precursor-basedsyntheses of high-pressure phases.

In addition to their innovations discussed above, the inventors haveunexpectedly discovered that the new compound Na₄Si₂₄ is an excellentprecursor in the production of another new compound Si₂₄.

In one embodiment of the invention, the Na₄Si₂₄ compound may besubjected to dynamic vacuum conditions at modest temperatures to removethe sodium atoms and produce a sodium-free version of the compound: Si₂₄(also named oC24 silicon or Cmcm-24 silicon).

Si₂₄, never before discovered or described in a publication, has anorthorhombic structure. The lattice constants are as follows: a=3.83 Å,b=10.69 Å and c=12.63 Å. This material is semi-conducting with anindirect band gap of 1.30 eV and a direct band gap of 1.34 eV. Si₂₄ is asignificantly better absorber of sunlight than d-Si.

The Si₂₄ structure of silicon is produced by treating the Na₄Si₂₄precursor under vacuum conditions (˜1×10⁻² torr) at elevated temperature(˜130° C.) for a preferred period of several days. At conditions of only80° C. and no vacuum (ambient pressure), sodium atoms start to leave theNa₄Si₂₄ structure. At 1×10⁻⁵ torr and 130° C. for 4 days, no sodium isdetectable by energy dispersive spectroscopy measurements (EDS).

As shown in FIG. 12A, reduced sodium content in the structure wasverified by x-ray diffraction. The top pattern in FIG. 12A shows thex-ray diffraction of Na₄Si₂₄ and provides the lattice constantspreviously outlined—i.e., a=4.106, b=10.563, c=12.243 with a sodiumoccupancy of 1.0. The bottom pattern in FIG. 12A shows the sodium-freeversion, Si₂₄, with lattice constants a=3.83, b=10.69, c=12.63 and Naoccupancy of 0.0.

The sodium content was also verified by using energy dispersivespectroscopy (EDS) on a scanning electron microscope (SEM).

FIG. 13 shows the electrical resistance as a function of temperature(1/T) and is fit to extract a band gap of 1.3 eV.

Density functional theory calculations were performed to confirm thestability of the Si₂₄ structure and to further investigate theelectronic band structure.

The light absorption of Si₂₄ relevant to solar applications was alsocalculated. FIG. 14 shows the absorption (imaginary portion of thedielectric constant) for Si₂₄ compared with “normal” diamond phasesilicon. Enhanced light absorption was observed compared with normalsilicon.

The studies discussed herein have been motivated by the potential tofind new silicon allotropes with advanced optical and electronicproperties beyond those of d-Si [B4, B8, B16-B18]. In particular,photovoltaic applications ideally require a direct band gap of ˜1.3 eV[B6], which has not been achieved by any existing silicon phase.Theoretically, low-energy silicon allotrope candidates were suggestedthat exhibit greatly improved visible light absorption characteristicswith quasidirect band gaps (nearly degenerate indirect and direct gaps)[B4, B8]; however, no experimental synthesis was reported thus far. Ofthe known metastable Si allotropes, the BC8 structure is likelysemi-metallic [B17] and the R8 structure was calculated to possess asmall indirect gap of 0.24 eV [B16]. Lonsdaleite silicon, produced byheating the BC8 structure above 470 K, has an indirect gap of ˜1 eV[B16, B17] and the crystal structure of alto-Si is not clearly resolved[B19]. Type-II silicon clathrate, Si₁₃₆, possesses a wide band-gap of1.9 eV [B18], which is not suitable for photovoltaic applications. Inaddition, symmetry analysis of the cubic Si₁₃₆ structure shows thatelectric dipole transitions associated with this gap are forbidden[B20].

Silicon-rich compounds may be considered as another route forsynthesizing novel classes of silicon allotropes. This approach was usedpreviously for the synthesis of type II silicon clathrate [B13] andgermanium clathrate [B19], both of which utilize chemical precursorsthat are formed at ambient pressure. In this discovery, applicantsconsidered using compounds recovered from high-pressure conditions aschemical precursors, rather than using compounds formed at ambientpressure. In this case, the synthesis of entirely new, previouslyinaccessible phases become possible by performing ambient-pressurechemical manipulations on inherently metastable materials recovered fromhigh pressure. Applicants disclose herein the formation of Na₄Si₂₄ above˜8 GPa, and the concomitant metastable recovery of this phase to ambientconditions [B5]. This compound consists of a channel-like sp³ siliconhost structure filled with linear Na chains as a guest structure. Theseopen channels that host Na suggest a possible pathway for Na removal viadiffusion along the channels as schematically shown in FIG. 15A.

By exposing recovered Na₄Si₂₄ samples to elevated temperatures, removalof Na from the structure was observed. This process occurs attemperatures as low as 320 K, while type-II silicon clathrates(Na_(x)Si₁₃₆) require much higher temperatures (>623 K) for Na removal[B13, B21]. Thermal “degassing” of Na₄Si₂₄ at 400 K under dynamic vacuumresulted in a gradual reduction of the Na content and Na was completelyremoved from structure over a period of eight days. FIG. 12A showspowder X-ray diffraction (PXRD) patterns obtained from Na₄Si₂₄ and froma sample recovered after the thermal “degassing” process. After theeight-day period the host structure remains unchanged (Cmcm); however,the lattice constants and PXRD peak intensities are significantlydifferent. The best Rietveld fits were obtained when sodium was excludedfrom the refinement (including Na led to unphysical negative occupanciesand thermal parameters), indicating that the sodium concentration isbelow the detectable limits using PXRD. Excellent agreement between theexperimental and calculated lattice constants, for both full and emptystructures, provides additional evidence for the formation of theNa-free Si₂₄ structure (FIG. 12C). By removing Na, the a, b, and clattice parameters change by −6.7%, +1.0%, and +2.9%, respectively,which indicates that the void channel diameters become larger and thelength along the channels is reduced upon Na removal. In addition, thecomputed Raman spectrum for the Na-free Si₂₄ structure is in excellentagreement with experiment, corroborating the absence of Na from the Si₂₄structure (discussed in more detail below).

The absence of sodium was also demonstrated using energy-dispersiveX-ray spectroscopy (EDXS). EDXS measurements were performed on“degassed” samples of Na_(x)Si₂₄ (0≦x≦4) (FIG. 12B). No sodium wasdetected in the EDXS spectrum for completely degassed samples,indicating that the sodium content is below the detection limit of theinstrument (<0.1 atom %). Thus, the new phase is at least 99.9% puresilicon and may be considered as a new allotrope: Si₂₄ (oC24).

Si₂₄ possesses an orthorhombic structure (Cmcm, space group 63) withlattice parameters a=3.825(1) Å, b=10.700(2) Å, and c=12.648(2) Å. Thereare three non-equivalent Si positions (discussed in more detail below)and each Si atom is connected tetrahedrally with bond lengths rangingfrom 2.33 to 2.41 Å, as compared with the bond length of d-Si (2.35 Å).Along the a-axis, Si₂₄ possesses octagonal linear channels, which wereoccupied by Na in Na₄Si₂₄. To maintain the void space, the bond anglesare distorted in a range from 93.7-135.9°, deviated from the perfecttetrahedral angle (109.5°). The low density of Si₂₄ (2.16 g/cc), due toits high nanoporosity, is between that of d-Si (2.33 g/cc) and Si₁₃₆clathrate (2.15 g/cc).

The thermodynamic and dynamic stability of the new silicon phase wasinvestigated using first principles calculations. Total energycalculations using density functional theory (DFT) show that Si₂₄possesses a higher enthalpy than d-Si by 0.09 eV/atom and isenergetically more favorable than other known metastable BC8 and R8phases [B4]. Phonon dispersion relations for Si₂₄, obtained from latticedynamics calculations (Supplementary Information, see below), indicatethat this structure is dynamically stable at both high- and low-pressureconditions. This fact demonstrates that the removal of sodium atoms fromNa₄Si₂₄ does not affect the lattice stability of the Si framework. Atambient pressure, the Si₂₄ lattice maintains dynamic stability, which isconsistent with our experimental observations at ambient conditions. Inour calculations, Si₂₄ is destabilized above 10 GPa and we speculatethat it might transform to the metallic Si-II (β-tin) structure, similarto type-II Si clathrate, above 12 GPa [B22].

To gain further insight into the properties of Si₂₄, the electronic bandstructure was calculated. Using DFT, Si₂₄ was calculated to have adirect band gap (E_(d)) of 0.57 eV and an indirect band gap (E_(i)) 0.53eV (Supplementary Information, see below). The difference between E_(d)and E_(i) is small, albeit formally an indirect band gap material. Thehighest valence and the lowest conduction bands are very flat in the 1-Zdirection, indicating a quasidirect gap nature for Si₂₄. It is awell-known limitation of standard DFT to underestimate the band gap ofsilicon. Therefore, we used quasiparticle (G₀W₀) calculations foraccurate band gap estimations. Under this approach, we successfullyreproduced the indirect gap value of 1.12 eV for d-Si (1.17 eV fromexperiment) and found that the G₀W₀ corrected E_(d) and E_(i) for Si₂₄are 1.34 eV and 1.30 eV, respectively (FIG. 16A, 16B). It is worthnoting that the indirect gap nature of Si₂₄ can be easily tuned to aformally direct gap material by uniaxial compression. As the conductionband minimum is located at the Z point and the valence band maximum isat the f point, our calculations show that a two percent latticecompression along the c-axis induces an indirect-to-direct gaptransition (Supplementary Information, see below).

The temperature dependence of the electrical conductivity σ, for Si₂₄,is shown in FIG. 16C. Si₂₄ exhibits semiconducting behavior where σincreases with increasing T. This is in contrast with the temperaturedependence of σ for Na₄Si₂₄ [B5], which is metallic due to the excesscharge carriers associated with Na. The rigid-band model, applicable toother guest-host compounds [B23], can be also applied to Na_(x)Si₂₄(0≦x≦4). According to this model, the electropositive Na donates itsvalence electrons to the conduction bands of the Si framework. Si₂₄, ascompared to Na₄Si₂₄, has no available conduction charge carriers andtherefore exhibits semiconducting behavior (FIG. 16C). Using DFT, wecalculated the electronic structure of Na_(x)Si₂₄ down to x=0.125 (˜0.5atom %) and found that this small Na content is enough to maintain themetallic nature of the compound (Supplementary Information, see below).Thus, a metal-to-semiconductor transition is observed when sodium iscompletely removed from the structure. A band gap of 1.3 eV wascalculated for Si₂₄ from the intrinsic region of the electricalconductivity (FIG. 16C), which is in excellent agreement with our G₀W₀calculations.

Optical reflectivity measurements were performed on samples of Si₂₄ inorder to obtain absorption information from the powder samples and tofurther evaluate the band gap. FIG. 16D shows Tauc plots [B24] of theKubelka-Munk absorption (K/S) with the data scaled for both indirect anddirect transitions. If a plot of (K/S·hv)^(1/2) versus hv yields astraight line, E_(i) can be estimated by extrapolating this line toK/S=0. Similarly, E_(g) is estimated from a plot of (K/S·hv)² versus hv.Under this approach, absorption edges for Si₂₄ were observed at 1.29 eVand 1.39 eV, assuming indirect and direct transitions, respectively.While single crystal or thin film samples should be measured for a morestrict discussion of the band gap, the observed absorption edges are inexcellent agreement with the energy gaps determined from electricalconductivity measurements and from theoretical estimates using firstprinciples calculations (Supplementary Information, see below). Thesevarious methods constrain the band gap of Si₂₄ near 1.3 eV and indicateonly minor separation between the direct and indirect values: 0.04 eVfrom the G₀W₀ calculation, ˜0.1 eV from experiment.

To check for potential improvements in light absorption properties, wecalculated absorption spectra of Si₂₄ as shown in FIG. 17, compared withd-Si, by solving the Bethe-Salpeter Equation (BSE) [B25, B26]. Theabsorption of these two phases is compared with the reference air mass(AM) 1.5 solar spectral irradiance [B27]. Compared with d-Si, the lightabsorption of Si₂₄ is significantly enhanced below 3 eV, particularly inthe visible light range where solar spectral irradiation has its maximumintensity. While the electric dipole transitions at the band gap ofSi₁₃₆ are forbidden by symmetry [B20], optical absorption in Si₂₄starting from the absorption edge is dipole allowed. It is interestingto note that the calculated absorption intensity of Si₂₄ isquantitatively comparable to that of ternary chalcopyrite semiconductorcompounds (CuInSe₂, CuGaSe₂) [B28], which are well-known thin-film solarcell materials [B4].

Applicants have presented the discovery of a new allotrope of silicon,Si₂₄, formed through a novel high-pressure precursor process. The sodiumconcentration was found to be <0.1 atom %, the laboratory detectionlimit. Electrical resistivity and optical reflectivity measurementsindicate that Si₂₄ is a semiconductor with a band gap of ˜1.3 eV, inexcellent agreement with our first principles calculations. Thedifference between E_(d) and E_(i) is negligibly small (<0.1 eV) andboth are well within the optical band gaps for photovoltaic applications(<1.5 eV), which is a unique property of this new silicon allotrope. Itis interesting that the band gap of Si₂₄ coincides with thetheoretically proposed optimal value that maximizes solar conversionefficiency for a single p-n junction to 33.7%, namely the‘Shockley-Queisser limit’ [B6]. Therefore, Si₂₄ appears to be apromising candidate for thin-film solar applications, which should beinvestigated further along with other properties such as carriermobility and potential for light emission. The quasidirect nature of theband gap allows for greatly improved optical properties, while thematerial maintains advantages of silicon, e.g., potential for doping,oxide layer, etc. The synthesis of Si₂₄ currently requires ahigh-pressure precursor, which places limitations on scalability foreventual applications. However, low-pressure methods such as chemicalvapor deposition could enable larger scale production of Si₂₄, as is thecase for diamond [B29], another high-pressure phase. Furthermore, theunique nanoporous nature of this structure may be of interest for gasand/or lithium storage and for molecular-scale filtering applications.More broadly, Si₂₄ expands the known allotropy in element fourteen andthe novel high-pressure precursor synthesis approach suggests potentialfor entirely new materials with desirable properties.

EXAMPLES Synthesis

Si₂₄ was synthesized in a two-step process. In the first step, Na₄Si₂₄was synthesized from a Na/Si mixture with 15 mol % Na. The mixture wasground in a ceramic mortar for one hour inside a high-purity Ar gloveboxand loaded into a Ta capsule. The capsule was then introduced in a 14/8multianvil assembly using a Re heater and ZrO₂ insulation. Shortingbetween Ta and Re was prevented by employing Al₂O₃ tubes, and a W—ReC-type thermocouple, imbedded in an Al₂O₃ plug, was used for accuratetemperature control. The mixture was pressurized in a 1500 tonmultianvil press at a rate of 10 bar/hour (oil pressure) to a pressureof 10 GPa and reacted at 800° C. in two steps: preheating at 400° C. for30 min, in order to avoid a blow-out of the overheated Na, and reactionat the final temperature for one hour, after which the sample wasquenched by switching off the power. The recovered sample was easilyremoved from the Ta capsule and washed with distilled water. Theresulting product of the reaction was polycrystalline Na₄Si₂₄. In thesecond step, polycrystalline agglomerates of Na₄Si₂₄ were placed in afurnace under a dynamic vacuum of ˜10⁻⁵ Torr and “degassed” at 125° C.for 8 days in order to obtain the empty Si₂₄ structure, which wassubsequently washed thoroughly with water.

Powder X-Ray Diffraction

PXRD data were collected on Rigaku Rapid diffractometer with MoKradiation and curved area detector. The sample to detector distance wasrefined using a high purity silicon standard. Rietveld refinements werecarried out using GSAS with EXPGUI software.

Electron Microscopy

EDXS measurements were performed using JEOL JSM-6500F microscopeequipped with Oxford Instruments X-max detector (80 mm²) and the datawere analyzed using the Aztec software.

Electrical Measurements

Electrical resistivity was measured with a Physical Property MeasurementSystem (PPMS) from Quantum Design using a two-probe method. Platinumwires (5 μm) were attached to the dense polycrystalline specimens (˜50μm in size) using Leitsilber conductive silver cement (Ted Pella, silvercontent 45%, sheet resistance: 0.02-0.04 ohms/square).

Optical Reflectivity

Optical reflectivity measurements were performed on polycrystallinepowder samples of Si₂₄ using the near/mid IR light source from anAgilent Cary 670 spectrometer. Reflected light was focused into adispersive spectrometer with CCD detector. A PTFE standard was used as areflectance reference. Reflectivity data were processed under theKubelka-Munk formalism and band gaps were estimated from Tauc plots.

First-Principles Calculations

For accurate band gap estimations, we have employed quasi-particlecalculations (GW) and a hybrid functional approach (HSE06) for acomparison. We used Bethe-Salpeter equation (BSE) to compute the Coulombcorrelation between the photoexcited electrons and holes. The fulldetails of first principles calculations, with complete references, canbe found in the Supplementary Information.

SUPPLEMENTARY INFORMATION

Electronic Structure Calculations

Electronic structure calculations and ionic relaxation were performedusing Density Functional Theory (DFT) [C1, C2] with the GeneralizedGradient Approximation (GGA) and Perdew, Burke, and Ernzerhof (PBE)exchange-correlation functional [C3, C4], as implemented in the QuantumESPRESSO software [C5]. Applicants used a plane-wave basis set cutoff of60 Ry and a Brillouin-zone integration grid of a 16×16×16 k-points.

Crystal Structure

TABLE 4 Crystallographic data for Si₂₄. Si₂₄ - full profile refinement,MoK_(a) Space group Cmcm (#63) a (Å)  3.8246(5) b (Å) 10.7002(18) c (Å)12.6476(19) Atomic coordinates x y z Occupancy U_(iso) Si1 (8f) 00.2427(8) 0.5553(5) 1 0.0176(11) Si2 (8f) 0 0.5718(6) 0.3439(6) 10.0176(11) Si3 (8f) 0 0.0295(6) 0.5918(5) 1 0.0176(11) Refinementstatistics X² = 0.3889 · 10⁻² wRp = 0.0859 (-Bknd) Rp = 0.0486 (-Bknd)

TABLE 5 Crystallographic data for Si₂₄ (DFT, PBE) Si₂₄ - DFT, PBEcalculations, 1 atm Space group Cmcm (#63) a (Å)  3.8475 b (Å) 10.7443 c(Å) 12.7342 Atomic coordinates x y z Occupancy Si1 (8f) 0 0.242850.55476 1 Si2 (8f) 0 0.57130 0.34274 1 Si3 (8f) 0 0.02862 0.59056 1Band-Gap and Absorption Calculations

Applicants calculated band gaps for d-Si and Si₂₄ using severalcomputational approaches to make it clear that Si₂₄ is a quasidirectband gap semiconductor. It is well-known issue that standard DensityFunctional Theory (PBE here) underestimates the band gap of materials.GW (where G means the single-particle Green's function and W thescreened Coulomb potential) calculations were performed to correct thePBE band gap values and the Bethe-Salpeter equation (BSE) [C6, C7] wasused to compute the Coulomb correlation between the photoexcitedelectrons and holes using the ABINIT software [C8]. Applicants conductedGW₀ calculations with the cutoff dielectric matrix of 5 Hartree, whichwas tested to various semiconductors and insulators successfully [C9].Applicants applied BSE calculations to d-Si for testing convergence ofthe calculations and then calculated Si₂₄. For BSE calculations,applicants used a cutoff of 3.0 Hartree for the dielectric matrix.

The GW approximation was applied to the self-energy Σ[C10, C11, C12](the proper exchange-correlation potential acting on an excited electronor hole), which can be written as the product of the one-electronGreen's function times the screened Coulomb interaction Σ=iGW. In ourcalculations, applicants have used both single shot GW(G₀W₀) andpartially self-consistent GW₀. It is worth noting that full correctionto both G and W (GW) on d-Si overestimated the band gap significantly[C13]. As shown in Table 6, for d-Si, G₀W₀ and GW₀ give excellentagreement with the experimental band gap for d-Si (1.17 eV) and in themain text, applicants used G₀W₀ results. The Heyd-Scuseria-Ernzershof(HSE) exchange-correlation functional [C14] was also tested by us, whichis more accurate for large band gap materials.

TABLE 6 Calculated band gaps for d-Si and Si₂₄ using variousfunctionals. Units are in eV. d-Si, Si_(24,) indirect indirect (direct)PBE 0.62^(b) 0.53 (0.57) HSE06 1.28^(a) 1.41 (1.45) G₀W₀ 1.12 1.30(1.34) GW₀ 1.2^(b) 1.43 (1.46) ^(a)Ref [C15], ^(b)Ref [C9]Lattice Parameter Change with Respect to Sodium Concentration

Applicants calculated the lattice parameters, a, b, and c for Na_(x)Si₂₄(0≦x≦4) at different values of x. Supercells of of Si₂₄ unit cells wereconstructed with only one sodium atom: 1x1x1(x=0.25), 2x1x1 (x=0.125),3x1x1 (x=0.083). Atomic positions were relaxed to determine theinfluence of Na content on the lattice parameters. Theoreticaloptimizations are in excellent agreement with experimental data forNa₄Si₂₄ and Si₂₄ (FIG. 18).

Metal-Insulator Transition in Na_(x)Si₂₄

Computationally, applicants checked if Na_(x)Si₂₄ becomes asemiconductor at low values of x. FIG. 19 shows the electronic densityof states of Na₁Si₂₄, Na_(0.5)Si₂₄, Na_(0.333)Si₂₄, and Na_(0.125)Si₂₄.By lowering sodium concentration, the system remains metallic and thereis no significant change in the number of conduction electrons at theFermi level.

Phonon Dispersion Relations

Phonon calculations were performed using Density Functional PerturbationTheory [C16], as implemented in the Quantum-espresso package. Theelectronic wave function was expanded with a kinetic energy cutoff of 60Ry. A Uniform with a 6×6×6 q-point mesh of phonon momentum has beencalculated with a 12×12×12 k-point mesh.

The dynamical stability of Si₂₄ was examined at ambient pressure and at10 GPa. FIG. 20 shows the evolution of the phonon dispersion relationsalong high symmetry lines and the corresponding phonon density ofstates. One can see that Si₂₄ is dynamically stable to 10 GPa. Abovethis pressure, it becomes destabilized, indicating a structuraltransformation.

Raman Scattering

Raman scattering data were collected from “degassed” samples Si₂₄. A 532nm diode laser was used as an excitation source and focused onto thesample using a 20× long working distance objective lens. The power atthe sample was approximately 10 mW. Scattered radiation was collected inthe 180° back-scatter geometry and focused onto a 50 μm confocalpinhole, which served as a spatial filter. This light was then passedthrough two narrow-band notch filters (Ondax, SureBlock) and focusedonto the entrance slit of a spectrograph (Princeton Instruments,SP2750). Light was dispersed off of an 1800 gr/mm grating and recordedusing a liquid nitrogen-cooled charge-coupled device detector (PrincetonInstruments, Plyon).

The Raman active mode was calcualted using density functionalperturbation theory [C17]. A Brlliouin zone sampling grid with 2π×0.04Å⁻¹ was used with a plane basis set cutoff of 500 eV. The ionicpositions were carefully relaxed at ambient pressure. FIG. 21 comparesexperimental and theoretical Raman data for Si₂₄.

Uniaxial Compression Effect on the Band Gap

Strictly speaking, Si₂₄ is an indirect band gap material, however,electronic dispersion relations show nearly flat bands along the f to Zdirection. Due to the small difference between E_(d) and E_(i),applicants examined band gap changes during uniaxial compression of Si₂₄along c-axis. The difference between the direct and indirect band gaps(E_(d)−E_(i)) is shown in FIG. 22. An indirect-to-direct band gaptransition occurs when the initial lattice constant, c₀, is reduced by˜2%.

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As the present invention may be embodied in several forms withoutdeparting from the spirit or essential characteristics thereof, it willbe understood that the invention is not limited by the details of theforegoing description, unless otherwise specified, but rather should beconstrued broadly within its spirit and scope as defined in the appendedclaims, and therefore all changes and modifications that fall within themetes and bounds of the claims. Accordingly, the invention is defined bythe appended claims.

The invention claimed is:
 1. A method of producing Si₂₄ comprising:forming an Na₄Si₂₄ precursor by reacting a mixture of silicon and sodiumat a pressure from about 7 GPa to about 15 GPa and a temperature fromabout 320° C. to about 1500° C.; subjecting the Na₄Si₂₄ precursor tovacuum conditions at a temperature from about 40° C. to about 500° C. toproduce Si₂₄.
 2. The method of claim 1, wherein the precursor Na₄Si₂₄ isdynamically stable at a temperature of about 25° C. and a pressure ofabout 1 atmosphere.
 3. The method of claim 1 wherein the Na₄Si₂₄precursor contains 24 silicon atoms and 4 sodium atoms, has a spacegroup of Cmcm, and has the following lattice constants: a=4.106 Å,b=10.563 Å and c=12.243 Å.
 4. The method of claim 1 wherein Si₂₄ has aquasi-direct band gap, with a direct gap value of 1.34 eV and anindirect gap value of 1.3 eV.
 5. The method of claim 1 wherein the ratioof sodium to silicon used to form Na₄Si₂₄ is from about 10 to about 30mol %.
 6. The method of claim 1 wherein the mixture of sodium andsilicon is heated in two steps, first to about 400° C. for about 30minutes and second to about 800° C. for about 1.5 to 24 hours.
 7. Amethod of producing Na₄Si₂₄ by reacting a mixture of silicon and sodiumat a pressure of greater than about 8 GPa and a temperature of greaterthan about 700° C.
 8. The method of claim 7 wherein Na₄Si₂₄ contains 24silicon atoms and 4 sodium atoms, has a space group of Cmcm, and has thefollowing lattice constants: a=4.106 Å, b=10.563 Å and c=12.243 Å. 9.The method of claim 7 wherein the ratio of sodium to silicon in thereaction is about 20 mol %.
 10. A compound of the formula Si_(2.4)having an orthorhombic structure.
 11. The compound of claim 10 whereinthe compound, has a space group of Cmcm (#63).
 12. The compound of claim11 having the following lattice constants: a=3.83 Å, b=10.69 Å andc=12.63 Å.
 13. The compound of claim 10 having an indirect band gap of1.3 eV and a direct band gap of 1.34 eV.
 14. A compound of the formulaNa₄Si₂₄.
 15. The compound of claim 14 which is dynamically stable at atemperature of about 25° C. and a pressure of about 1 atmosphere. 16.The compound of claim 14 wherein Na₄Si₂₄ contains 24 silicon atoms and 4sodium atoms; has a space group of Cmcm; and has the following latticeconstants: a=4.106 Å, b=10.563 Å and c=12.243 Å.